A Dunkl Analogue of Operators Including Two-variable Hermite polynomials
aa r X i v : . [ m a t h . C A ] A p r A Dunkl Analogue of Operators Including Two-variableHermite polynomials
Rabia Akta¸s, Bayram C¸ ekim, and Fatma Ta¸sdelen
Abstract.
The aim of this paper is to introduce a Dunkl generalization ofthe operators including two variable Hermite polynomials which are definedby Krech [ ] (Krech, G. A note on some positive linear operators associatedwith the Hermite polynomials, Carpathian J. Math., 32 (1) ( ) , 71–77)and to investigate approximating properties for these operators by means ofthe classical modulus of continuity, second modulus of continuity and Peetre’sK-functional.
1. Introduction
Up to now, linear positive operators and their approximation properties havebeen studied by many research workers, see for example [ ], [ ], [ ], [ ], [ ], [ ],[ ] and references therein. Also, linear positive operators defined via generatingfunctions and their further extensions are intensively studied by a large number ofauthors. For various extensions and further properties, we refer for example Altinet.al [ ], Dogru et. al [ ], Olgun et.al [ ], Sucu et. al [ ], Tasdelen et.al [ ],Varma et.al [
24, 25 ].Recently, linear positive operators generated by a Dunkl generalization of theexponential function have been stated by many authors. In [ ], Dunkl analogueof Sz´asz operators by using Dunkl analogue of exponential function was given asfollows S ∗ n ( g ; x ) = 1 e ν ( nx ) ∞ X k =0 ( nx ) k γ ν ( k ) g (cid:18) k + 2 νθ k n (cid:19) ; n ∈ N , ν, x ∈ [0 , ∞ ) (1.1)for g ∈ C [0 , ∞ ) where Dunkl analogue of exponential function is defined by e ν ( x ) = ∞ X k =0 x k γ ν ( k ) (1.2)for k ∈ N and ν > − and the coefficients γ ν are as follows γ ν (2 k ) = 2 k k !Γ ( k + ν + 1 / ν + 1 /
2) and γ ν (2 k + 1) = 2 k +1 k !Γ ( k + ν + 3 / ν + 1 /
2) (1.3)
Mathematics Subject Classification.
Primary 41A25, 41A36; Secondary 33C45.
Key words and phrases.
Dunkl analogue, Hermite polynomial, modulus of continuity, Ko-rovkin’s type approximation theorem. in [ ]. Also, the coefficients γ ν verify the recursion relation γ ν ( k + 1) γ ν ( k ) = (2 νθ k +1 + k + 1) , k ∈ N , (1.4)where θ k = (cid:26) , if k = 2 p , if k = 2 p + 1 (1.5)for p ∈ N ∪ { } . Similarly, Stancu-type generalization of Dunkl analogue of Sz´asz-Kantorovich operators and Dunkl generalization of Sz´asz operators via q-calculushave been defined in [
11, 12 ] and for other research see [
15, 16 ].The two-variable Hermite Kampe de Feriet polynomials H n ( ξ, α ) are definedby (see [ ]) ∞ X n =0 H n ( ξ, α ) n ! t n = e ξt + αt from which, it follows H n ( ξ, α ) = n ! [ n ] X k =0 α k ξ n − k k !( n − k )! . In a recent paper, Krech [ ] has introduced the class of operators G αn givenby G αn ( f ; x ) = e − ( nx + αx ) ∞ X k =0 x k k ! H k ( n, α ) f (cid:18) kn (cid:19) , x ∈ R +0 , n ∈ N , α ≥ G αn .In the present paper, we first give the Dunkl generalization of two variableHermite polynomials and then we define a class of operators by using the Dunklgeneralization of two variable Hermite polynomials. We give the rates of conver-gence of the operators T n to f by means of the classical modulus of continuity,second modulus of continuity and Peetre’s K -functional and in terms of the ele-ments of the Lipschitz class Lip M ( α ) .
2. The Dunkl generalization of two variable Hermite polynomials
The Dunkl generalization of two variable Hermite polynomials is defined by ∞ X n =0 H µn ( ξ, α ) n ! t n = e αt e µ ( ξt ) (2.1)from which, we conclude H µn ( ξ, α ) = n ! [ n ] X k =0 α k ξ n − k k ! γ µ ( n − k ) , which gives the two variable Hermite polynomials as µ = 0 . For our purpose, wedenote h µn ( ξ, α ) = γ µ ( n ) H µn ( ξ, α ) n ! DUNKL ANALOGUE OF OPERATORS INCLUDING TWO-VARIABLE HERMITE POLYNOMIALS3 and we can write that the polynomials h µn ( ξ, α ) are generated by ∞ X n =0 h µn ( ξ, α ) γ µ ( n ) t n = e αt e µ ( ξt ) (2.2)where h µn ( ξ, α ) = γ µ ( n ) [ n ] X k =0 α k ξ n − k k ! γ µ ( n − k ) . In order to obtain some properties of h µn ( ξ, α ) , we remind the following definitionand lemma given in [ ]. Definition . [ ] Let µ ∈ C ( C := C \ (cid:8) − , − , ... (cid:9) , x ∈ C and let ϕ beentire function. The linear operator D µ is defined on all entire functions ϕ on C by D µ ( ϕ ( x )) = ϕ ′ ( x ) + µx ( ϕ ( x ) − ϕ ( − x )) , x ∈ C . (2.3) We use the notation D µ,x since D µ is acting on functions of the variable x . Thus, D µ,x ( ϕ ( x )) = ( D µ ϕ ) ( x ) . Lemma . [ ] Let ϕ, ψ be entire functions. For the linear operator D µ , thefollowing statements hold i ) D jµ : x n → γ µ ( n ) γ µ ( n − j ) x n − j , j = 0 , , , ..., n ( n ∈ N ); D jµ : 1 → ii ) D µ ( ϕψ ) = D µ ( ϕ ) ψ + ϕ D µ ( ψ ) , where ϕ is an even function iii ) D µ : e µ ( λx ) → λe µ ( λx ) . By using these definition and lemma, we can state the next result.
Lemma . For the Dunkl generalization of two variable Hermite polynomials h µn ( ξ, α ) , the following results hold true ( i ) ∞ X n =0 h µn +1 ( ξ,α ) γ µ ( n ) t n = ( ξ + 2 αt ) e αt e µ ( ξt )( ii ) ∞ X n =0 h µn +2 ( ξ,α ) γ µ ( n ) t n = ( ξ + 4 ξαt + 4 α t + 2 α ) e αt e µ ( ξt ) + 4 αµe αt e µ ( − ξt ) Proof.
Applying the linear operator D µ in view of Lemma 1 , we have D µ ( te µ ( ξt )) = ( tξ + 1) e µ ( ξt ) + 2 µe µ ( − ξt ) D µ ( e αt ) = 2 αte αt . (2.4)Also applying the linear operator D µ to both side of the generating function(2.2), we have ∞ X n =0 h µn ( ξ, α ) γ µ ( n ) D µ ( t n ) = D µ ( e αt e µ ( ξt )) . By using (2.4) and Lemma 1 (i), we get the first relation . Similarly, if we apply thelinear operator D µ to the relation in (i), we get ∞ X n =0 h µn +1 ( ξ, α ) γ µ ( n ) D µ ( t n ) = D µ h ( ξ + 2 αt ) e αt e µ ( ξt ) i RABIA AKTAS¸, BAYRAM C¸ EKIM, AND FATMA TAS¸DELEN from (2.4) and Lemma 1, it follows ∞ X n =0 h µn +2 ( ξ, α ) γ µ ( n ) t n = ( ξ + 4 ξαt + 4 α t + 2 α ) e αt e µ ( ξt ) + 4 αµe αt e µ ( − ξt ) . (cid:3) Definition . With the help of the Dunkl generalization of two variable Her-mite polynomials given in (2.2), we introduce the operators T n ( f ; x ) , n ∈ N givenby T n ( f ; x ) := 1 e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k f (cid:18) k + 2 µθ k n (cid:19) (2.5) where α ≥ , µ ≥ and x ∈ [0 , ∞ ) . The operators (2.5) are linear and positive. Inthe case of µ = 0 , it gives G αn given by (1.6) Lemma . For the operators T n ( f ; x ) , we can obtain the following equations: ( i ) T n (1; x ) = 1( ii ) T n ( t ; x ) = x + αx n ( iii ) T n ( t ; x ) = x + αn x + αn x + α n x + xn + µxn e µ ( − nx ) e µ ( nx ) Proof.
By using the generating function in (2.2), the relation ( i ) holds. Forthe proof of ( ii ) , in view of the recursion relation in (1.4), we get T n ( t ; x ) = 1 ne αx e µ ( nx ) ∞ X k =1 h µk ( n, α ) γ µ ( k − x k . When we replace k by k + 1, we obtain (ii) by use of Lemma 2 (i). For the proofof ( iii ) , by using (1.4) , we have T n ( t ; x ) = xn e αx e µ ( nx ) ∞ X k =0 ( k + 1 + 2 µθ k +1 ) h µk +1 ( n, α ) γ µ ( k ) x k . From the equation θ k +1 = θ k + ( − k , (2.6)it yields T n ( t ; x ) = xn e αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k ) h µk +1 ( n, α ) γ µ ( k ) x k + xn e αx e µ ( nx ) ∞ X k =0 (1 + 2 µ ( − k ) h µk +1 ( n, α ) γ µ ( k ) x k . Using the recursion relation in (1.4) in the first series, it follows T n ( t ; x ) = x n e αx e µ ( nx ) ∞ X k =0 h µk +2 ( n, α ) γ µ ( k ) x k + xn e αx e µ ( nx ) ∞ X k =0 h µk +1 ( n, α ) γ µ ( k ) x k + 2 µxn e αx e µ ( nx ) ∞ X k =0 ( − x ) k h µk +1 ( n, α ) γ µ ( k )from Lemma 2 (i) and (ii), we complete the proof of (iii). (cid:3) DUNKL ANALOGUE OF OPERATORS INCLUDING TWO-VARIABLE HERMITE POLYNOMIALS5
Lemma . As a consequence of Lemma 3, we can give the next results for T n operators ∆ = T n ( t − x ; x ) = 2 αx n ∆ = T n (( t − x ) ; x ) = 1 n x (cid:0) x α + 4 αx + n (cid:1) + 2 µxn e µ ( − nx ) e µ ( nx ) (2.7) Theorem . For T n operators and any uniformly continuous bounded function g on the interval [0 , ∞ ) , we can give T n ( g ; x ) uniformly ⇒ g ( x ) on each compact set A ⊂ [0 , ∞ ) when n → ∞ . Proof.
From Korovkin Theorem in [ ], when n → ∞ , we have T n ( g ; x ) uniformly ⇒ g ( x ) on A ⊂ [0 , ∞ ) which is each compact set because lim n →∞ T n ( e i ; x ) = x i , for i = 0 , , , which is uniformly on A ⊂ [0 , ∞ ) with the help of using Lemma4. (cid:3) Theorem . The operator T n maps C B ( R +0 ) into C B ( R +0 ) and k T n ( f ) k ≤ k f k for each f ∈ C B ( R +0 ) .
3. Convergence of operators in (2.5)
In what follows we give some rates of convergence of the operators T n . Firstly,we recall some definitions as follows. Let Lip M ( α ) Lipschitz class of order α. If g ∈ Lip M ( α ), the inequality | g ( s ) − g ( t ) | ≤ M | s − t | α holds where s, t ∈ [0 , ∞ ) , < α ≤ M > . e C [0 , ∞ ) is the space of uniformlycontinuous on [0 , ∞ ) . The modulus of continuity g ∈ e C [0 , ∞ ) is denoted by ω ( g ; δ ) := sup s,t ∈ [0 , ∞ ) | s − t |≤ δ | g ( s ) − g ( t ) | . (3.1)We first estimate the rates of convergence of the operators T n by using modulus ofcontinuity and in terms of the elements of the Lipschitz class Lip M ( α ) . Theorem . If h ∈ Lip M ( α ) , we have | T n ( h ; x ) − h ( x ) | ≤ M (∆ ) α/ where ∆ is given in Lemma 4. Proof.
Since h ∈ Lip M ( α ), it follows from linearity | T n ( h ; x ) − h ( x ) | ≤ T n ( | h ( t ) − h ( x ) | ; x ) ≤ M T n ( | t − x | α ; x ) . From Lemma 4 and H¨older’s famous inequality, we can write | T n ( h ; x ) − h ( x ) | ≤ M [∆ ] α . Thus, we find the required inequality. (cid:3)
RABIA AKTAS¸, BAYRAM C¸ EKIM, AND FATMA TAS¸DELEN
Theorem . The operators in (2.5) verify the inequality | T n ( g ; x ) − g ( x ) | ≤ s n x (4 x α + 4 xα + n ) + 2 µx e µ ( − nx ) e µ ( nx ) ! ω (cid:18) g ; 1 √ n (cid:19) , where g ∈ e C [0 , ∞ ) . Proof.
By Lemma 4, from Cauchy-Schwarz’s inequality and the property ofmodulus of continuity | g ( t ) − g ( x ) | ≤ w ( g ; δ ) (cid:18) | t − x | δ + 1 (cid:19) , (3.2)it follows | T n ( g ; x ) − g ( x ) | ≤ T n ( | g ( t ) − g ( x ) | ; x ) ≤ (cid:18) δ T n ( | t − x | ; x ) (cid:19) ω ( g ; δ ) ≤ (cid:18) δ p ∆ (cid:19) ω ( g ; δ ) . Then from Lemma 4, one has | T n ( g ; x ) − g ( x ) | ≤ δ s n x (4 x α + 4 αx + n ) + 2 µxn e µ ( − nx ) e µ ( nx ) ! ω ( g ; δ ) , (3.3)by choosing δ = √ n , we completes the proof. (cid:3) Let C B [0 , ∞ ) denote the space of uniformly continuous and bounded functionson [0 , ∞ ). Also C B [0 , ∞ ) = { g ∈ C B [0 , ∞ ) : g ′ , g ′′ ∈ C B [0 , ∞ ) } (3.4)with the norm k g k C B [0 , ∞ ) = k g k C B [0 , ∞ ) + k g ′ k C B [0 , ∞ ) + k g ′′ k C B [0 , ∞ ) for ∀ g ∈ C B [0 , ∞ ) . Lemma . For h ∈ C B [0 , ∞ ) , the following inequality holds true | T n ( h ; x ) − h ( x ) | ≤ [∆ + ∆ ] k h k C B [0 , ∞ ) , (3.5) where ∆ and ∆ are given by in Lemma 4. Proof.
From the Taylor’s series of the function h , h ( s ) = h ( x ) + ( s − x ) h ′ ( x ) + ( s − x ) h ′′ ( ̺ ) , ̺ ∈ ( x, s ) . Applying the operator T n to both sides of this equality and then using the linearityof the operator, we have T n ( h ; x ) − h ( x ) = h ′ ( x ) ∆ + h ′′ ( ̺ )2 ∆ . DUNKL ANALOGUE OF OPERATORS INCLUDING TWO-VARIABLE HERMITE POLYNOMIALS7
From Lemma 4, it yields | T n ( h ; x ) − h ( x ) | ≤ αx n k h ′ k C B [0 , ∞ ) + (cid:20) n x (cid:0) x α + 4 αx + n (cid:1) + 2 µxn e µ ( − nx ) e µ ( nx ) (cid:21) k h ′′ k C B [0 , ∞ ) ≤ [∆ + ∆ ] k h k C B [0 , ∞ ) , which finishes the proof. (cid:3) Now we recall that the second order of modulus continuity of f on C B [0 , ∞ ) isgiven as ω ( f ; δ ) := sup . Here M is a positive constant. Now, we can give the importanttheorem. Theorem . For the operators by (2.5), the following inequality holds | T n ( g ; x ) − g ( x ) | ≤ M ( min (cid:18) , χ n ( x )2 (cid:19) k g k C B [0 , ∞ ) + ω g ; r χ n ( x )2 !) (3.8) where ∀ g ∈ C B [0 , ∞ ) , x ∈ [0 , ∞ ) , M is a positive constant which is independent of n and χ n ( x ) = ∆ + ∆ . Proof.
For any f ∈ C B [0 , ∞ ), from the triangle inequality, we can writeΘ = | T n ( g ; x ) − g ( x ) | ≤ | T n ( g − f ; x ) | + | T n ( f ; x ) − f ( x ) | + | g ( x ) − f ( x ) | from Lemma 5, which followsΘ ≤ k g − f k C B [0 , ∞ ) + χ n ( x ) k f k C B [0 , ∞ ) = 2 n k g − f k C B [0 , ∞ ) + χ n x ) k f k C B [0 , ∞ ) o . From (3.6), we have Θ ≤ K (cid:18) g ; χ n ( x )2 (cid:19) , which holdsΘ ≤ M ( min (cid:18) , χ n ( x )2 (cid:19) k g k C B [0 , ∞ ) + ω g ; r χ n ( x )2 !) from (3.7). (cid:3) Similar to the proof of above theorem, simple computations give the next the-orem.
RABIA AKTAS¸, BAYRAM C¸ EKIM, AND FATMA TAS¸DELEN
Theorem . If g ∈ C B [0 , ∞ ) and x ∈ [0 , ∞ ) , we get | T n ( g ; x ) − g ( x ) | ≤ M ω g ; 12 s n x (8 x α + 4 xα + n ) + 2 µxn e µ ( − nx ) e µ ( nx ) ! + ω (cid:18) g ; 2 αx n (cid:19) where M is a positive constant. Remark . The case of µ = 0 in Theorem 6 gives the result given in [ ] . References [1] Altın, A., Do˘gru, O., Ta¸sdelen, F. The generalization of Meyer-K¨onig and Zeller operatorsby generating functions, J. Math. Anal. Appl., 312 (1) (2005) , 181-194.[2] Altomare, F., Campiti, M. Korovkin-Type Approximation Theory and its Applications, deGruyter Studies in Mathematics, vol. 17, Walter de Gruyter, Berlin, Germany, .[3] Appell, P., Kampe de Feriet, J. Hypergeometriques et Hyperspheriques: Polynomesd’Hermite, Gauthier-Villars, Paris, 1926.[4] Atakut, C¸ ., ˙Ispir, N. Approximation by modified Sz´asz–Mirakjan operators on weightedspaces, Proc. Indian Acad. Sci. Math. 112 (2002) , 571–578[5] Atakut, C¸ ., B¨uy¨ukyazici, ˙I. Stancu type generalization of the Favard Sz´asz operators, Appl.Math. Lett., 23 (12) (2010) , 1479-1482.[6] Ciupa, A. A class of integral Favard–Sz´asz type operators. Stud. Univ. Babes-Bolyai Math.40 (1) (1995) , 39–47.[7] DeVore, R.A., Lorentz, G.G. Construtive Approximation, Springer, Berlin, .[8] Do˘gru, O., ¨Ozarslan, M.A., Ta¸sdelen, F. On positive operators involving a certain class ofgenerating functions, Studia Sci. Math. Hungar., 41 (4) (2004) , 415-429.[9] Gadzhiev, A.D. The convergence problem for a sequence of positive linear operators on un-bounded sets and theorems analogues to that of P.P. Korovkin, Sov. Math. Dokl. 15 (5) (1974) , 1453-1436.[10] Gupta, V., Vasishtha, V., Gupta, M.K. Rate of convergence of the Sz´asz–Kantorovich–Bezieroperators for bounded variation functions, Publ. Inst. Math. (Beograd) (N.S.), 72 (2006) ,137–143.[11] ˙Icoz, G., Cekim, B. Dunkl generalization of Sz´asz operators via q-calculus, Journal of In-equalities and Applications, 2015:284 (2015 ), 11 pages.[12] ˙Icoz, G., Cekim, B. Stancu-type generalization of Dunkl analogue of Sz´asz–Kantorovich op-erators, Math. Meth. Appl. Sci, 39 (2016) , 1803–1810.[13] Korovkin, P. P. On convergence of linear positive operators in the space of continuous func-tions (Russian), Doklady Akad. Nauk. SSSR (NS) 90 (1953) , 961–964.[14] Krech, G. A note on some positive linear operators associated with the Hermite polynomials,Carpathian J. Math., 32 (1) (2016) , 71–77.[15] Mursaleen, M., Rahman, S., Alotaibi. A. Dunkl generalization of q-Sz´asz-Mirakjan Kan-torovich operators which preserve some test functions, Journal of Inequalities and Applica-tions, 2016:317 (2016) , 18 pages.[16] Mursaleen, M., Nasiruzzaman, Md., Alotaibi. A.On modified Dunkl generalization of Sz´aszoperators via q-calculus, Journal of Inequalities and Applications, 2017:38 (2017),
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